Gallelian Relativity Part 2 ( Reference Frames)

Mathematics Requires a Small dose, not of Genius but of Imaginative Freedom, which in larger dose would be Insanity..

Angus K. Rodgers

In the previous lesson we have learnt the basics of Gallelian Relativity.. And we've also took a look at those very foundation upon which the whole subject is built.. In the upcoming lessons we'll try to exploit this theory in it's full glory..

Today's topic is

Reference Frame

In the previous lesson we've dealt with the definition of the very notion.. So, in summary it's just a consistent way to describe the implicit properties of a body or system..

Before starting I would like to remind you that in here we'll only talk about the Euclidean Space (Flat Space), Space which strictly follows these 7 axioms..

1. A point is that which has no part.
2. A line is breadthless length.
3. The ends of a line are points.
4. A straight line is a line which lies evenly with the points on itself.
5. A surface is that which has length and breadth only.
6. The edges of a surface are lines.
7. A plane surface is a surface which lies
evenly with the straight lines on itself.

These axioms were put forward by Euclid of Alexandria in his famous book Elements which laid the foundations of Mathematics of nearly two millennia..

Euclid of Alexandria

Elements

Although the axioms look pretty convincing and matches our daily perceptions till in extreme cases it falls apart..

An Example of Non Euclidean Geometry 

More explicit general ideas of Geometry like Non Euclidean one was introduced and perfected by 18th and 19th century's Mathematicians like Lobachevsky, Gauss, Riemann and others.. But for our consideration we'll stick with the Euclidean Geometry..

Nikolai Lobachevsky 

Carl Friedrich Gauss 

Bernhard Riemann 

Reference Frame in Euclidean Space 


Now if I ask you how to do you describe the position of an object from your perspective explicitly??

Well, the short answer which you might've guessed is to assign an unique number which represents the distance of the object from you.. And you are correct..

If you are a being that dwells on a line it's the only way you can describe the position of the very object..

But things start to get a little murky once start you live on a plane, you'll see only the distance between you and the object isn't sufficient to accurately describe the position..

Because when you were an one dimensional being or a being from the line you could move only one way either you could move forward or backward..but you couldn't move right or left..
But now since you live on a plane, you can move left-right along with forward backward..

So, we can clearly conclude that only one number is not enough to unambiguously describe the position of the object on a plane.. we need another number..

And In three-dimensional space, the space where an object can have motion towards up down along with these other two, we need to introduce another number..

But what will the new number depict??
Well it turns out it depends on how you choose to describe the position..

Here's a short note in Mathematics or physics if we need to take account of the direction to precisely describe a point or motion, then the point or motion is known as a Vector.. we use a letter with an arrow on top of it to describe a vector.. The arrow is the unique identity of the vector..

And to depict the space in we use capital R with some exponent on top, the exponent defines the number of dimensions..

So for example if we write vector V ∈ R³ then it signifies that our vector V is located on a three-dimensional Space..

Another point to be remembered that to define a n dimensional space we need n AXES all perpendicular to each other..

In 3 dimensional coordinate system we need 3 axes each perpendicular to each other..

Besides there various Coordinate Systems which follows specific transformation rules, each of them have some unique advantages over the others on a particular topic.. Among them there are especially 2 Coordinate systems which are handy in your daily experiences..



Polar/Spherical Coordinate System:-

If you consider a fixed line as your stationary line, then as always the first number will point out the distance and the second number will depict the angle(θ), which the direct line from the object towards you make with the stationary line..

This particular type of description is called the POLAR COORDINATE SYSTEM..

From this picture we can clearly say the point is 2 unit away from the origin and
π/4 radians away from our stationary line..


The three dimensional modification of this Coordinate System is known as Spherical Coordinate System..
As usual we need another number to consider this another degree of freedom..

Conventionality the plane slicing the space horizontally is known as Azimuth reference and the plane orthogonal to the Azimuthal plane is known as Zenith reference..

And the three numbers (r,θ,Φ) define the Euclidean distance, the inclination angle(angle between the zenith direction or Z direction and line OP), the azimuth or Azimuthal angle(angle between azimuth direction and the orthogonal projection on Azimuthal plane, ON) respectively..



Cartesian Coordinate System (Rectangular Coordinate System)

In this specific type of defining space on a plane involves considering two lines perpendicular to each other, these lines are called AXES.. This method was first unveiled by philosopher René Descartes in the 16th Century..In this method the position of any object is defined by the consecutive distance from the axes..

René Descartes 

Traditionally we use the symbol î for the first axis or X axis and ĵ for y axis to define a unit length along these directions..

 In this following context the point P is located at the place where it is 3 units is away from x axis and 2 units away from Y axis..

For 3 dimensional space we need another number to incorporate the distance of the point from Z direction and we use k̂ to define an unit vector at Z direction..

Interchange between the two Coordinate Systems

To build a somewhat a sophisticated formulation we need to workout a general transformation rules between the Coordinate system..

Polar to Cartesian
If the mutually perpendicular axes on Azimuthal plane are X and Y consecutively and the concerned point or vector make the Azimuthal angle and inclination angle φ and θ respectively where the zenith line is Z axis, then the transformation equation is the following

 Here the rows on the spherical side depicts the Radial distance, Inclination angle and Azimuth consecutively and the rows on the Cartesian side refers to the X axis, Y axis and Z axis from top to bottom..

Cartesian to Polar
If the Coordinate of the Vector on Cartesian system is (x,y,z) then the corresponding coordinate on spherical system can be expressed by the following relation..

 On the next part we'll explore the transformation rules in Galleian Relativity and we'll explore its consequences..